Time travel is possible, according to physicists. The math of relativity allows it.
However, time travel is impractical and unfeasible due to the very relativity that makes it possible. Feasible and possible are not the same thing in this case.
Let's reason it through.
Let's say you invent and manufacture a fully functional time machine. Inebriated by your own awesomeness at accomplishing this formidible feat and compelled to action by hubris, you get in and with a surprising lack of your usual forethought, you turn some dials and send yourself one year into the past. A moment later you find yourself in the cold vaccuum of interstellar space and promptly die as the bubble of air that time traveled with you rapidly dissipates.
You see, your machine worked perfectly. The problem is, a year prior to your ill fated leap through time, the space occupied by you and your time machine was occupied by...well...space. Empty space.
In remarkable, albeit short-lived, 20/20 hindsight, you realize that your choice to go one year in the past had a certain very narrowly defined logic to it. It takes the earth just about a year to go around the sun and all else in the universe held constant, you might have arrived only about six hours before the arrival of the prior year's Earth in that vicinity of its orbital plane around the Sun, give or take a few thousand miles. When the earth finally arrived, you frozen dessicated corpse would hit the atmosphere and incinerate.
Even if you had designed your time machine to compensate for the Earth's roughly 365.25 day transit around the Sun (derp!), your cosmic provincialism would still likely have been your undoing. In a year's local time, the entire solar system would have moved a fair piece due to the rotation of the Milky Way galaxy, never mind the exponential expansion of the universe itself.
Now let's reset the clock, if you will pardon the expression. Assume you thought of all these technical obstacles in advance and gave your time machine the computing power to extrapolate the movement of space across past and future time (let's set aside the increasing margins of error the farther in time you decide to go...think 10 day weather forecasts). You still have another issue.
After years of R&D, prototype testing, and countless deaths of lab rats and rhesus monkeys (you can't say you miss the monkeys...ornery little buggers), you finally turn on your fully functional time machine. Depending on your ultimate design, one of two things is possible.
The most probable design is that your time machine transports through time whatever is placed WITHIN it, but does not transport the time machine itself. This is the safest time machine design (though not for rats and monkeys). If you could transport the machine itself, there would be no need for all those years of writing grants and toiling over your PhD thesis in quantum physics. You could just transport back in time to visit your penniless graduate student self and hand over all the research, schematics, and calculations so he or she could make short work of it. Your past self would seem overjoyed and promptly take you out for celebratory beers. At some point in the evening, your past self drugs and murders your future self and then hides your machine in a secure storage facility while pretending to have a highly successful career as a research physicist, eventually bringing forth the time machine you worked so hard on and winning the Nobel Prize you should have won. Was it homicide? Or suicide?
Now let's return from that alternate universe where the diabolically evil you resides. The present good you, seeking to avoid ethical conundrums, goes with prototype A, the one that only transports whatever is within its transportation field. You design in some quantum computing so that anything transported through time can only reappear within the time field of another time machine similar or identical in design to your own. Using quantum communication, your machine can identify accessible points in spacetime, which gives you kind of a weird creepy feeling.
If you identify accessible points in spacetime prior to the first functional activation of your machine, prepare for trouble and give up any notion of free will.
Another option is to transport a smaller time machine using the first time machine. This second time machine is programmed with only one destination, to return you to the inside of the first time machine. Dangers here include malfunctions of the first or second time machine. Watch the show "Sliders" for a crude representation of the dangers inherent in this design. If the second smaller machine always returns you to the exact point in spacetime when you left, you probably don't have to worry about time machine #1 malfunctioning before your return. You'll want time machine 2 to be very well manufactured and preferably airtight and well stocked with food, water, and oxygen.